Time-mapping estimates Directory UMM :Journals:Journal_of_mathematics:EJQTDE:

3 Time-mapping and auxiliary functions

3.1 Time-mapping estimates

Let’s consider the initial value problem I =    −ϕ p u ′ ′ = gu on R u0 = s, u ′ 0 = 0 Where g : R → R is a continuous function satisfying sgnsgs −c for c 0 and Gs → +∞. The function τ g defined by τ g s = 2c p sgns Z s dξ [Gs − Gξ] 1p for s in R with Gs = R s gξ dξ, c p = 1 p ∗1p and p ∗ = p p−1 is the time-mapping associated to I. Under the assumptions sgnsgs −c for c 0 and Gs → +∞ when |s| → +∞, τ g s is well defined for |s| large enough. By adapting arguments developed in [12] for the case p = 2, one can easily derive that for s large enough I admits a periodic solution u s with ||u s || ∞ = s and τ g s is the value of the half period. Time-mapping enables to provide a-priori estimates for solutions of boundary value problems cf [7], [10], [11], [12]. Here, we give new results on the time-mapping estimates extending and even improving some results in [7], [10], [11], [12]. Lemma 3.1. Assume that there exist positive real numbers k ± and k ± such that lim sup s→±∞ pGs|s| p = k ± resp. lim inf s→±∞ pGs|s| p = k ± then lim inf s→±∞ τ g s ≥ π p k ± 1p resp. lim sup s→±∞ τ g s ≤ π p k ± 1p Proof. One can notice that under the assumption sgnsgs −c for c 0 and the fact that k ± k ± are greater than 0, Gs → +∞ when |s| → +∞ so that τ g s is well defined for s large enough. Let’s limit the proof of the lemma to the cases lim sup s→−∞ pGs|s| p = k − , resp. lim inf s→−∞ pGs|s| p = k − , the other cases being similar. EJQTDE, 2009 No. 57, p. 10 For s 0 and for any ξ such that |s| p |ξ| p , we have lim sup s→−∞ pGs |s| p − |ξ| p = lim sup s→−∞ pGs|s| p × lim s→−∞ |s| p |s| p − |ξ| p = k − and then lim sup s→−∞ pGs |s| p − |ξ| p − pGξ |s| p − |ξ| p = k − . So for ǫ 0, there is a real number s 0 such that for s s we have Gs − Gξ ≤ 1pk − + ǫ|s| p − |ξ| p . Recalling the expression of τ g s and taking into account inequality above, we get τ g s ≥ 2c p Z s s dξ [Gs − Gξ] 1p ≥ 2c p p 1p k − + ǫ 1p Z s s dξ [|s| p − |ξ| p ] 1p . Setting z = ξs, one has τ g s ≥ 2p − 1 1p k − + ǫ 1p Z 1 dz [1 − z p ] 1p = π p k − + ǫ 1p for all s s 0. Thus lim inf s→−∞ τ s ≥ π p k − 1p . For the case lim inf s→−∞ pGs|s| p = k − , we have ∀ǫ 0, there is a real number s 0 such that for s s Gs − Gξ ≥ 1pk − − ǫ|s| p − |ξ| p , for all s ξ 0. So, for ǫ sufficiently small such that k − − ǫ 0, we have τ g s ≤ 2c p p 1p k − − ǫ 1p Z s s dξ [|s| p − |ξ| p ] 1p . And by a simple computation as previously done we get lim sup s→−∞ τ g s ≤ π p k − 1p . EJQTDE, 2009 No. 57, p. 11

3.2 Auxiliary functions related to the time-mapping